A238208 The total number of 1's in all partitions of n into an odd number of distinct parts.

0, 1, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 24, 28, 33, 38, 45, 52, 60, 69, 80, 91, 105, 120, 137, 156, 178, 202, 230, 261, 295, 334, 378, 426, 481, 542, 609, 685, 769, 862, 966, 1082, 1209, 1351, 1508, 1681, 1873, 2086, 2319, 2578

Offset

0,9

Comments

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

Or: the number of partitions of n-1 into an even number of distinct parts >=2. - R. J. Mathar, May 11 2016

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Andrew Howroyd)

Formula

a(n) = Sum_{j=1..round(n/2)} A067661(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067659(n-2*j).

G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) + (1/2)*(x/(1-x))*(Product_{n>=1} 1 - x^n).

a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 17 2020

From Peter Bala, Feb 02 2021: (Start)

a(n+1) = d(n) - ( d(n-1) + d(n-3) ) + ( d(n-4) + d(n-6) + d(n-8) ) - ( d(n-9) + d(n-11) + d(n-13) + d(n-15) ) + ( d(n-16) + d(n-18) + d(n-20) + d(n-22) + d(n-24) ) - ( d(n-25) + d(n-27) + d(n-29) + d(n-31) + d(n-33) + d(n-35) ) + ..., where d(n) = A000009(n) is the number of partitions of n into distinct parts, with the convention that d(n) = 0 for n < 0.

G.f.: x/(1 - x^2)*Sum_{n >= 0} (-1)^n*x^((n^2+n+1-(-1)^n)/2)/Product_{k = 1..n} 1 - x^k.

Alternative g.f.: ( Product_{k >= 1} 1 + x^k ) * x*Sum_{n >= 0} (-1)^n*x^(n^2)*(1 - x^(2*n+2))/(1 - x^2).

Faster converging g.f. (conjecture): Sum_{n >= 0} x^((n+1)*(2*n+1))/ Product_{k = 1..2*n} 1 - x^k. (End)

Example

a(10) = 3 because the partitions in question are: 7+2+1, 6+3+1, 5+4+1.

Maple

A238208 := proc(n)
    local a, L, Lset;
    a := 0 ;
    L := combinat[firstpart](n) ;
    while true do
        # check that parts are distinct
        Lset := convert(L, set) ;
        if nops(L) = nops(Lset) then
            # check that number is odd
            if type(nops(L), 'odd') then
                if 1 in Lset then
                    a := a+1 ;
                end if;
            end if;
        end if;
        L := combinat[nextpart](L) ;
        if L = FAIL then
            return a;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 11 2016
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
    end:
a:= n-> b(n-1, 2, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, May 01 2020

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i+1, t] + b[n-i, i+1, 1-t]]];
a[n_] := b[n-1, 2, 1];
a /@ Range[0, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

Prog

(PARI) seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) + eta(x + A)/(1-x))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

Crossrefs

Column k=1 of A238450.

Cf. A067659, A067661, A133280.

Sequence in context: A185327 A210717 A171962 * A350892 A029028 A240572

Adjacent sequences: A238205 A238206 A238207 * A238209 A238210 A238211

Keyword

nonn,easy

Author

Mircea Merca, Feb 20 2014

Extensions

a(51)-a(60) from R. J. Mathar, May 11 2016

Status

approved